Hegel’s Science of Logic

# Quantity

## Chapter 3: The Quantitative Relation or Quantitative Ratio

§ 669

The infinity of quantum has been determined to the stage where it is the negative beyond of quantum, which beyond, however, is contained within the quantum itself. This beyond is the qualitative moment as such. The infinite quantum as the unity of both moments, of the quantitative and qualitative determinateness, is in the first instance a ratio.

§ 670

In the ratio, quantum no longer has merely an indifferent determinateness but is qualitatively determined as simply related to its beyond. It continues itself into its beyond; this, in the first place, is simply another quantum. But they are essentially related to each other not as external quanta, but the determinateness of each consists in this relation to the other. In this their otherness they have thus returned into themselves; what each is, it is in the other; the other constitutes the determinateness of each. The flight of quantum away from and beyond itself has now therefore this meaning, that it changed not merely into an other, or into its abstract other, into its negative beyond, but that in this other it reached its determinateness, finding itself in its beyond, which is another quantum. The quality of quantum, the specific nature of its Notion, is its externality as such, and in ratio the quantum is now posited as having its determinateness in its externality, in another quantum, and as being in its beyond what it is.

§ 671

They are quanta which stand to each other in the relation just described. This relation is itself also a magnitude; the quantum is not only in a ratio, but it is itself posited as a ratio; there is only a single quantum and this has the said qualitative determinateness within itself. As such ratio, it is a self-enclosed totality and indifferent to limit, and it expresses this by containing within itself the externality of its determining and by being in this externality related only to itself. It is therefore, in its own self infinite.

§ 672

Ratio as such is:

1. direct ratio. In this, the qualitative moment does not yet emerge explicitly as such; its mode is still only that of quantum, namely, to be posited as having its determinateness in its very externality. The quantitative ratio is in itself the contradiction of externality and self-relation, of the affirmative being of quanta and their negation; next, it is sublated;
2. in the indirect or inverse ratio, in which is posited the negation of one of the quanta with the alteration of the other, and the alterableness of the direct relation itself;
3. in the ratio of powers, however, the unity which in its difference is self-related, vindicates itself as a simple self-production of the quantum; this qualitative moment itself, when finally posited in a simple determination and as identical with the quantum, becomes measure.

§ 673

About the nature of the following ratios, much has been anticipated in the preceding Remarks concerning the infinite of quantity, i.e., the qualitative moment in it; it only remains therefore to expound the abstract Notion of these ratios.

### A. THE DIRECT RATIO

§ 674

1. In the ratio which, as immediate, is direct, the determinateness of either quantum lies reciprocally in the determinateness of the other. There is only one determinateness or limit of both and this is itself a quantum, namely, the exponent of the ratio.

§ 675

2. The exponent is any quantum; but it is self-related in its own externality and a qualitatively determined quantum only in so far as it has within itself its own difference, its beyond and its otherness. This difference of quantum present within it is, however, the difference of unit and amount, unit being a being which is determined as for-itself, and amount the indifferent fluctuation of the determinateness, the external indifference of quantum. Unit and amount were at first moments of quantum; now, in the ratio, in the realisation so far of quantum, each of its moments appears as a quantum on its own, and as a determination of its existence - as a limiting of the otherwise merely external, indifferent determinateness of quantity.

§ 676

The exponent is this difference as a simple determinateness, i.e. it has immediately within it the significance of both determinations. It is first of all a quantum. As such it is amount; if the one side of the ratio which is taken as unit is expressed numerically as one-and it counts only as such-then the other, the amount, is the quantum of the exponent itself. Secondly, it is simple determinateness as the qualitative moment of the sides of the ratio. When the quantum of one side is determined, the other, too, is determined by the exponent and it is quite immaterial how the first is determined; it no longer has any significance as a determinate quantum on its own, but can equally well be any other quantum without altering the value of the ratio, which depends solely on the exponent. The one which is taken as unit always remains unit however great it becomes, and the other, no matter how great it, too, becomes in consequence, must remain the same amount of that unit.

§ 677

3. The two therefore constitute strictly only one quantum; the one has relatively to the other only the value of unit, not of amount, the other only that of amount; consequently, according to the specific nature of their Notion, they themselves are not complete quanta. But this incompleteness is a negation in them and a negation, not as regards their alterableness generally in virtue of which one of them (and it can be either of the two) can assume any possible magnitude, but as regards the determination that when one is altered the other is increased or diminished by the same amount; this means, as has been shown, that the quantum of only one of them, the unit, is altered, while the other side, the amount, remains the same quantum of units; but the former too still counts only as a unit no matter how it is altered as quantum. Thus each side is only one of the two moments of quantum and the independence which belongs to the peculiar character of quantum is in principle negated; in this qualitative relationship they are to be posited as negative relatively to each other.

§ 678

The exponent ought to be the complete quantum, since the determination of both sides coincides in it; but in fact, even as quotient it has the value only of amount or of unit. There is nothing to determine which side of the ratio must be taken as unit or which as amount; if the quantum B is measured in terms of quantum A as unit, then the quotient C is the amount of such units; but if A is itself taken as amount, the quotient C is the unit which to the amount A is required for the quantum B. This quotient therefore is, as exponent, not posited as what it ought to be-the determinant of the ratio, or the ratio's qualitative unity. It is only posited as this in so far as it has the value of being the unity of both moments, of unit and amount. True, these sides are present as quanta as they should be in the explicated quantum, in the ratio; but at the same time they have only the value proper to them as sides of the ratio, namely to be incomplete quanta and to count only as one of those qualitative moments. They must therefore be posited with this their negation and thus there arises a more developed form of the ratio, one which corresponds more to its character, a ratio in which the exponent has the significance of the product of the sides. As thus determined, it is the inverse ratio.

### B. INVERSE RATIO

§ 679

The ratio as now before us is the sublated direct relation. At first, the ratio was immediate and therefore not yet truly determinate; the determinateness it now possesses is such that the exponent counts as a product, as a unity of unit and amount. As we have already seen, the exponent as immediate could equally well be taken either as unit or as amount; it was then also only a simple quantum and therefore, by choice, an amount; one side was the unit, to be taken as a numerical one, of which the other side is a fixed amount and, at the same time, the exponent; the quality of this latter, therefore, was simply that this quantum is taken as fixed, or rather that this fixed quantum has the meaning only of quantum.

§ 680

Now in the inverse ratio the exponent as quantum is likewise immediate and is any quantum assumed as fixed. But this quantum is not a fixed amount to the one of the other quantum in the ratio; this ratio, which previously was fixed, is now on the contrary posited as alterable; if in place of the unit on one side of the ratio another quantum is taken, then the other side is no longer the same amount of units of the first side. In the direct ratio this unit is only the common element of both sides; as such, it continues itself into the other side, into the amount; and the amount itself taken by itself, or the exponent, is indifferent to the unit.

§ 681

But now the ratio is so determined that the amount as such is altered relatively to the other side of the ratio, to the unit; when another quantum is taken for the unit, the quantum of amount also is altered. Consequently, although the exponent is also only an immediate quantum arbitrarily assumed as fixed, it is not preserved as such in the side of the ratio, but this side, and with it the direct ratio of the sides, is alterable. In the ratio, then, as now before us, the exponent as the determining quantum is posited as negative towards itself as a quantum of the ratio, and hence as qualitative, as a limit-with the result that the qualitative moment is manifested independently and in distinct contrast to the quantitative moment. In the direct ratio, the alteration of the two sides is only the one alteration of the quantum which is taken as unit, the common element of both sides; by as many times as the one side is increased or diminished, so also is the other: the ratio itself is not affected by this alteration which is external to it. In the indirect ratio on the other hand, although the alteration, in keeping with the indifferent quantitative moment is also arbitrary, it is confined within the ratio, and this arbitrary quantitative fluctuation, too, is limited by the negative determinateness of the exponent as by a limit.

§ 682

2. We have now to consider more closely this qualitative nature of the inverse ratio, more particularly in its realisation, and to unravel the entanglement of the affirmative moment with the negative which is contained in it. The indirect or inverse ratio is quantum posited as a qualitative quantum, i.e., displaying itself as self-determining, as a limit of itself within itself. As such, the quantum is, first, an immediate magnitude as a simple determinateness, the whole as a quantum simply affirmatively present. But secondly, this immediate determinateness is also a limit; for that purpose the quantum is differentiated into two quanta which in the first instance are mutually related as others; but the quantum as their qualitative and, moreover, complete determinateness is the unity of unit and amount, a product of which the two quanta are the factors. Thus on the one hand the exponent of their ratio is in them identical with itself and is their affirmative moment which constitutes them quanta; on the other hand the exponent, as the negation posited in them, is the unity in them, so that although each is in the first place simply an immediate, limited quantum, it is at the same time limited in such a manner that it is only in principle [an sich] identical with its other. Thirdly, the exponent as the simple determinateness is the negative unity of this differentiation of itself into two quanta and is the limit of their reciprocal limiting.

§ 683

In conformity with these determinations, each of the two moments has its limit within the exponent, and since this is their specified unity each is the negative of the other; one of them becomes as many times smaller as the other becomes greater, the magnitude of each depending on its containing the magnitude which the other lacks. Each in this way continues itself negatively into the other; to the extent that each is amount, the amount of the other is cancelled, and each is what it is only through the negation or limit posited in it by the other. In this manner each also contains the other and is measured by it, for each is supposed to be only that quantum which the other is not; the magnitude of the other is an indispensable factor in the value of each and is therefore inseparable from it.

§ 684

This continuity of each in the other constitutes the moment of unity through which they are in ratio - the moment of the one determinateness, of the simple limit which is the exponent. This unity, the whole, constitutes the in-itself, the principle of each, from which their actual magnitude is distinct; in accordance with the latter, each, only is to the extent that it takes from the other a part of their common in-itself, the whole. But it can take from the other only as much as will make its own self equal to this in-itself; it has its maximum in the exponent which in accordance with the stated second determination is the limit of their reciprocal limiting. And since each is a moment of the ratio only in so far as it limits the other and is simultaneously limited by it, it loses this its determination in making itself equal to its in-itself; for in so doing not only does the other magnitude become zero, but it vanishes itself, since what it is supposed to be is not a mere quantum as such, but only quantum as such moment of a ratio. Thus each side is the contradiction between its determination as the in-itself, i.e. as unity of the whole, which is the exponent, and its determination as moment of the ratio; this contradiction is infinity again in a fresh, peculiar form.

§ 685

The exponent is a limit of the sides of its ratio within which they increase and decrease relatively to each other; but they cannot become equal to the exponent because of the latter's affirmative determinateness as quantum. As thus the limit of their reciprocal limiting, the exponent, is [a] their beyond, to which they infinitely approximate but which they cannot reach. This infinity in which they approach their beyond is the spurious infinity of the infinite progress; it is itself finite, is bounded by its opposite, by the finitude of each side and of the exponent itself and is consequently only approximation. But [b] the spurious infinity is here also posited as what it is in truth, namely, as only the negative moment in general, in accordance with which the exponent is the simple limit of the differentiated quanta of the ratio as their in-itself; their finitude, as their simple alterableness, is related to this in-itself which, however, remains absolutely distinct from them as their negation. This infinity, then, to which these quanta can only approximate is likewise affirmatively present and actual-the simple quantum of the exponent. In it is reached the beyond with which the sides of the ratio are burdened; it is implicitly the unity of both, and so implicitly the other side of each; for each has only as much value as the other has not, the whole determinateness of each thus resides in the other, and this their in-itself as an affirmative infinity is simply the exponent.

§ 686

3. The outcome of this, however, is the transition of the inverse ratio into a different determination from that which it had at first. This consisted in an immediate quantum being also related to another quantum in such a way that its increase is proportional to the decrease of the other, that it is what it is through a negative relationship with the other; also, a third magnitude is the common limit of this their fluctuating increase. This fluctuation here is their distinctive character - in contrast to the qualitative moment as a fixed limit; they have the character of variable magnitudes, for which the said fixed limit is an infinite beyond.

§ 687

But the determinations which have emerged and which we have to summarise are not only that this infinite beyond is at the same time some present finite quantum or other, but that its fixity - which constitutes it such infinite beyond relatively to the quantitative moment, and which is the qualitative moment of being only as abstract self-relation - has developed as a mediation of itself with itself in its other, in the finite terms of the ratio. The general result can be indicated by saying that the whole, as exponent, is the limit of the reciprocal limiting of both terms and is therefore posited as negation of the negation, hence as infinity, as an affirmative relation to itself. More specifically the exponent, simply as product, is implicitly the unity of unit and amount, but as each term is only one of these two moments, the exponent also includes them within itself and in them is implicitly related to itself. But in the inverse ratio, the difference has developed into the externality of quantitative being, and the qualitative moment is not merely the fixity of the exponent, or merely the immediate inclusion in it of the two moments of unit and amount, but is the identification of the exponent with itself in its self-external otherness. It is this determination which stands out as result in these moments as explicated. The exponent, namely, is found to be the in-itself which is realised in the simple alterableness of its moments as quanta; the indifference of their magnitudes in their alteration is displayed as an infinite progress, the basis of which is that in their indifference their determinateness is to have their value in the value of the other. Hence [a], in accordance with the affirmative aspect of the quantum, each is implicitly the whole of the exponent. Similarly [b], the quanta have for their negative moment, for their reciprocal limiting, the magnitude of the exponent; their limit is that of the exponent. The fact that they no longer have any other immanent limit, a fixed immediacy, finds expression in the infinite progress of their determinate being and of their limitation, in the negation of every particular value. This negation is, accordingly, the negation of the self-externality of the exponent which is displayed in the moments of the ratio; and the exponent, which is itself a simple quantum and is also differentiated into quanta, is, therefore, posited as preserving itself and uniting with itself in the negation of the indifferent existence of the quanta, thus being the determinant of its self-external otherness.

The ratio is now specified as the ratio of powers.

### C. THE RATIO OF POWERS

§ 688

1. The quantum which, in its otherness, is identical with itself and which determines the beyond of itself, has reached the stage of being-for-self. As a qualitative totality-for it posits itself as developed-it has for its moments the determinations of the Notion of number, unit and amount; in the inverse ratio, the latter is still a plurality determined not by the unit itself as such, but from elsewhere, by a Third; but now it is posited as determined only by the unit. This is the case in the ratio of powers where the unit, which in its own self is amount, is also amount relatively to itself as unit. The otherness, the amount of units, is the unit itself. The power is a plurality of units each of which is this same plurality. The quantum as an indifferent determinateness undergoes alteration; but in so far as this alteration is a raising to a power, this its otherness is limited purely by itself. Thus in the power, quantum is posited as returned into itself; it is at once its own self and also its otherness.

§ 689

The exponent of this ratio is no longer an immediate quantum as it is in the direct ratio and also in the inverse ratio. In the ratio of powers it is of a wholly qualitative nature-this simple determinateness that the amount is the unit itself, that the quantum in its otherness is identical with itself. In this is also contained the quantitative aspect of its nature, namely, that the limit or negation is not present simply affirmatively as an immediacy, but that the determinate being [of the quantum] is posited as continued into its otherness; for the truth of quality is just this, to be quantity, immediate determinateness as sublated.

§ 690

2. The ratio of powers appears at first to be an external alteration to which any quantum can be subjected; but it has a closer connection with the Notion of quantum: namely, that in the determinate being into which it has developed in the ratio of powers, quantum has reached its Notion and has completely realised it. This ratio is the display of what quantum is in itself and it expresses that determinateness or quality of quantum which is its distinctive feature. Quantum is the indifferent determinateness, i.e., posited as sublated, determinateness as a limit which is equally no limit, which continues itself into its otherness and so remains identical with itself therein. In the ratio of powers this quality of quantum is posited; quantum itself determines its otherness, its going beyond itself into another quantum.

§ 691

Comparing the progressive realisation of quantum in the preceding ratios, we see that the quality of quantum as the posited difference of itself from itself is simply this: to be a ratio. As a direct ratio it is at first only the simple or unmediated form of such posited difference, so that its self-relation which it has as exponent, in contrast to its differences, counts only as the fixity of an amount of the unit. In the inverse ratio, the quantum is negatively determined as relating itself to itself-to itself as a negation of itself in which, however, it has its value; as an affirmative relation to itself it is an exponent which, as quantum, is only in principle the determinant of its moments. But in the ratio of powers, quantum is present in the deference as its own difference from itself. The externality of the determinateness is the quality of quantum and this externality is now posited in conformity with the Notion of quantum, as the latter's own self-determining, as its relation to its own self, as its quality.

§ 692

3. But with the positing of quantum in conformity with its Notion, it has undergone transition into another determination; or, as we may also express it, its determination is now also a determinateness, what quantum is in principle it is now also in reality. It is quantum in so far as the externality or indifference of its determining (as it is said, it is that which can be increased or decreased) counts and is posited only simply or immediately; it has become the other of itself, namely, quality-in so far as this externality is now posited as mediated by quantum itself, and thus as moment of itself-so that in this very externality quantum is self-related, is being as quality.

§ 693

At first, then, quantity as such appears in opposition to quality; but quantity is itself a quality, a purely self-related determinateness distinct from the determinateness of its other, from quality as such. But quantity is not only a quality; it is the truth of quality itself, the latter having exhibited its own transition into quantity. Quantity, on the other hand, is in its truth the externality which is no longer indifferent but has returned into itself. It is thus quality itself in such a manner. that apart from this determination there would no longer be any quality as such. The positing of the totality requires the double transition, not only of the one determinateness into its other, but equally the transition of this other, its return, into the first. The first transition yields the identity of both, but at first only in itself or in principle; quality is contained in quantity, but this is still a one-sided determinateness. That the converse is equally true, namely, that quantity is contained in quality and is equally only a sublated determinateness, this results from the second transition-the return into the first determinateness. This observation on the necessity of the double transition is of great importance throughout the whole compass of scientific method.

§ 694

Quantum is now no longer an indifferent or external determination but as such is sublated and is quality, and is that by virtue of which something is what it is; this is the truth of quantum, to be Measure.

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